On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

Abstract

AbstractWe study pseudospectral and spectral functions for Hamiltonian system and differential equation with matrix‐valued coefficients defined on an interval with the regular endpoint a. It is not assumed that the matrix weight is invertible a.e. on . In this case a pseudospectral function always exists, but the set of spectral functions may be empty. We obtain a parametrization of all pseudospectral and spectral functions σ by means of a Nevanlinna parameter τ and single out in terms of τ and boundary conditions the class of functions y for which the inverse Fourier transform converges uniformly. We also show that for scalar equation the set of spectral functions is not empty. This enables us to extend the Kats–Krein and Atkinson results for scalar Sturm–Liouville equation to such equations with arbitrary coefficients and and arbitrary non trivial weight .